Regularization effect on autoencoder

import time
from typing import Tuple, Dict, Union, List

import tensorflow as tf
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import numpy as np
from sklearn.manifold import TSNE
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_samples, silhouette_score

NUMBER_PARAM = 10
REGULARIZER_TYPE = 'kernel_regularizer'
LS_KERNEL_REGULARIZER = np.linspace(0, 1e-4, NUMBER_PARAM)
LS_ACTIVITY_REGULARIZER = np.zeros(NUMBER_PARAM)

1. Data prep

(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()

x_train_norm = x_train/np.max(x_train)
x_test_norm = x_test/np.max(x_test)

2. Model architecture

The following model is to minimize loss function, $L_T$, in terms of loss from regular autoencoder, $L$, plus regularization term, $R$.

Encoding layer:
$h = \alpha_e(W_1 \times\ x + b_1)$, where $\alpha_e(.)$ is activation function ReLU, and the number of hidden units in $h$ is 196. Hence $h$ is a 196x1 vector in 196-dim latent space, $W_1$ is 196x748 weight matrix and $b_1$ is the bias term in the form of 196x1 vector.
Decoding layer:
$x' = \alpha_d(W_2 \times\ h + b_2)$, where $\alpha_d(.)$ is activation function sigmoid, $x'$ is the output of the autoencoder which is optimized to reconstruct back to input $x$.
Loss function:
$L_T = L + R = ||x - x'||^2 + \lambda_a\sum |h_i| + \lambda_k\sum |W_1|$, where $i$ is the number of hidden units, $\lambda_k\ $and $\lambda_a\ $are kernel and activity regularizer respectively, l1 regularization is used.

class Autoencoder(tf.keras.Model):
    def __init__(
        self, 
        input_shape: Tuple[int,int], 
        encoding_dim: int,
        activity_regularizer: float,
        kernel_regularizer: float,
        **kwargs
    ) -> None:
        
        super().__init__(**kwargs)
        self.encoding_dim = encoding_dim
        self.encoder = tf.keras.Sequential([
            tf.keras.layers.Flatten(),
            tf.keras.layers.Dense(
                encoding_dim, 
                activation='relu', 
                activity_regularizer=tf.keras.regularizers.L1(activity_regularizer),
                kernel_regularizer=tf.keras.regularizers.L1(kernel_regularizer)
            )
        ])
        self.decoder = tf.keras.Sequential([
          tf.keras.layers.Dense(
              input_shape[0]*input_shape[1], 
              activation='sigmoid'
          ),
          tf.keras.layers.Reshape(input_shape)
        ])
        
    def call(self, inputs):
        encoded = self.encoder(inputs)
        decoded = self.decoder(encoded)
        return decoded

3. Training

def training(
    train_set: np.ndarray,
    test_set: np.ndarray,
    activity_regularizer: float,
    kernel_regularizer: float,
    encoding_dim: int,
) -> Dict[str, Union[Autoencoder, float, int, np.ndarray]]:
    
    # build
    autoencoder = Autoencoder(
        input_shape=(28, 28),
        encoding_dim=encoding_dim,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
    )
    autoencoder.compile(optimizer='adam', loss='mse')
    callback = tf.keras.callbacks.EarlyStopping(monitor='val_loss', patience=3, min_delta=0.0001)

    # fit
    print(f'Training start:')
    print(f'activity_regularizer = {activity_regularizer}')
    print(f'kernel_regularizer = {kernel_regularizer}')
    start = time.time()
    history = autoencoder.fit(
        train_set, train_set,
        callbacks=[callback],
        epochs=100,
        batch_size=256,
        shuffle=True,
        verbose=0,
        validation_data=(test_set, test_set)
    )
    end = time.time()

    # eval
    training_loss = history.history.get('loss')[-1]
    testing_loss = history.history.get('val_loss')[-1]
    epoch = max(history.epoch)
    print('Training results:')
    print(f'training_loss = {training_loss}')
    print(f'testing_loss = {testing_loss}')
    print(f'epoch = {epoch}')
    print(f'time passed = {int(round(end-start))}s')
    print('-'*100)
    
    # return
    w1, _, _, _ = autoencoder.get_weights()
    w1_reshape = w1.T.reshape((encoding_dim,28,28))
    result = {
        'hyperparam': {
            'activity_regularizer': activity_regularizer,
            'kernel_regularizer': kernel_regularizer,
        },
        'results': {
            'training_loss': training_loss, 
            'testing_loss': testing_loss, 
            'epoch': epoch,
            'model': autoencoder,
            'w1_reshape': w1_reshape,
        }
    }
    
    return result

training_results = list()
for kernel_regularizer, activity_regularizer in zip(LS_KERNEL_REGULARIZER,LS_ACTIVITY_REGULARIZER):
    result = training(
        train_set=x_train_norm, 
        test_set=x_test_norm,
        activity_regularizer=activity_regularizer,
        kernel_regularizer=kernel_regularizer,
        encoding_dim=196
    )
    training_results.append(result)

Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0
Training results:
training_loss = 0.001473878393881023
testing_loss = 0.0014665934722870588
epoch = 21
time passed = 30s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 1.1111111111111112e-05
Training results:
training_loss = 0.005007409490644932
testing_loss = 0.004947695415467024
epoch = 55
time passed = 72s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 2.2222222222222223e-05
Training results:
training_loss = 0.007151870522648096
testing_loss = 0.007050069980323315
epoch = 55
time passed = 71s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 3.3333333333333335e-05
Training results:
training_loss = 0.008163469843566418
testing_loss = 0.008001532405614853
epoch = 72
time passed = 91s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 4.4444444444444447e-05
Training results:
training_loss = 0.00946621410548687
testing_loss = 0.00927070714533329
epoch = 68
time passed = 86s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 5.555555555555556e-05
Training results:
training_loss = 0.010429237969219685
testing_loss = 0.010227708145976067
epoch = 77
time passed = 98s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 6.666666666666667e-05
Training results:
training_loss = 0.011244737543165684
testing_loss = 0.011030125431716442
epoch = 65
time passed = 80s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 7.777777777777778e-05
Training results:
training_loss = 0.012330700643360615
testing_loss = 0.012090915814042091
epoch = 72
time passed = 88s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 8.888888888888889e-05
Training results:
training_loss = 0.013071522116661072
testing_loss = 0.01280683558434248
epoch = 75
time passed = 92s
----------------------------------------------------------------------------------------------------
Training start:
activity_regularizer = 0.0
kernel_regularizer = 0.0001
Training results:
training_loss = 0.014319300651550293
testing_loss = 0.014037657529115677
epoch = 75
time passed = 92s
----------------------------------------------------------------------------------------------------

4. Loss plot

MSE loss ($L_T$) against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def loss_plot(results) -> None:
    param = list()
    training_loss = list()
    testing_loss = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        training_loss.append(result['results']['training_loss'])
        testing_loss.append(result['results']['testing_loss'])

    plt.figure(figsize=(12, 8))
    plt.plot(param, testing_loss, 'bs', label='testing_loss')
    plt.plot(param, training_loss, 'r^', label='training_loss')
    plt.ylabel('loss')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training loss vs testing loss')
    plt.legend(loc='upper left')
    plt.show()
    plt.close()

loss_plot(results=training_results)

5. Sparsity plot

Sparsity of $h$ against hyperparameter, kernel_regularizer ($\lambda_k$), for training set and testing set

def _sparsity_map(x: int) -> None:
    if x != 0:
        return 1
    else:
        return 0
    
_sparsity_map_vec = np.vectorize(_sparsity_map)

def plot_sparsity(
    results, 
    train_set, 
    test_set
) -> None:
    param = list()
    ls_training_sparsity = list()
    ls_testing_sparsity = list()
    for result in results:
        param.append(result['hyperparam'][REGULARIZER_TYPE])
        model = result['results']['model']
        train_encoded_imgs = model.encoder(train_set).numpy()
        test_encoded_imgs = model.encoder(test_set).numpy()
        train_sparsity = np.sum(_sparsity_map_vec(train_encoded_imgs))/train_encoded_imgs.size
        testing_sparsity = np.sum(_sparsity_map_vec(test_encoded_imgs))/test_encoded_imgs.size
        ls_training_sparsity.append(train_sparsity)
        ls_testing_sparsity.append(testing_sparsity)
        
    plt.figure(figsize=(12, 8))
    plt.plot(param, ls_testing_sparsity, 'bs', label='testing_sparsity')
    plt.plot(param, ls_training_sparsity, 'r^', label='training_sparsity')
    plt.ylabel('sparsity')
    plt.xlabel(REGULARIZER_TYPE)
    plt.title('Training sparsity vs testing sparsity')
    plt.legend(loc='upper right')
    plt.show()
    plt.close()

plot_sparsity(
    results=training_results,
    train_set=x_train_norm,
    test_set=x_test_norm,
)

6. Weight matrix of the encoder

Weight matrix of the encoder, $W_1$, is shown on a grey-scale heatmap. Each of the subplot showing a row frm $W_1$ reshaped to 28x28

def plot_w1(
    w1: np.ndarray
) -> None:
    
    w1_dim = int(np.sqrt(len(w1)))
    fig, ax = plt.subplots(
        nrows=w1_dim, 
        ncols=w1_dim, 
        figsize=(w1_dim,w1_dim)
    )
    plt.gray()

    i = 0
    for row in ax:
        for col in row:
            col.imshow(w1[i])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            i = i + 1
            
    plt.show()
    plt.close()

for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    plot_w1(
        w1=result['results']['w1_reshape']
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

7. Original image vs latent space vs reconstructed image

def plot_images(
    img_gp: List[np.ndarray],
    num_img: int,
) -> None:
    
    _, ax = plt.subplots(nrows=len(img_gp), ncols=num_img, figsize=(num_img,len(img_gp)))
    plt.gray()
    for i, row in enumerate(ax):
        tmp = img_gp[i]
        for j, col in enumerate(row):
            col.imshow(tmp[j])
            col.get_xaxis().set_visible(False)
            col.get_yaxis().set_visible(False)
            
    plt.show()
    plt.close()

num_img = 20
img = x_test_norm[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    decoded_img = model.decoder(encoded_img).numpy()
    latent_dim = int(np.sqrt(len(encoded_img.T)))
    encoded_img_reshape = encoded_img.reshape((num_img,latent_dim,latent_dim))
    norm_encoded_img_reshape = norm_encoded_img.reshape((num_img,latent_dim,latent_dim))
    
    img_gp = [img, encoded_img_reshape, norm_encoded_img_reshape, decoded_img]
    plot_images(
        img_gp=img_gp,
        num_img=num_img,
    )

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

8. Latent space similarity plot

def similarity_plot(
    img: np.ndarray, 
    ls_label: List[int]
) -> None:
    
    similarity_matrix = img @ img.T
    ls_similarity = [list() for i in range(10)]
    ls_disimilarity = [list() for i in range(10)]
    for i, u in enumerate(similarity_matrix):
        for j, v in enumerate(u):
            if i == j:
                continue
            if ls_label[i] == ls_label[j]:
                label = ls_label[i]
                ls_similarity[label].append(v)
            else:
                label = ls_label[i]
                ls_disimilarity[label].append(v)
    
    plot_data = [
        {
            'data': ls_similarity,
            'title': 'Cosine similarity of the latent vector with the same label'
        },
        {
            'data': ls_disimilarity,
            'title': 'Cosine similarity of the latent vector with the different label',
        }
    ]
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(24,8))
    for idx, col in enumerate(ax):
        col.boxplot(plot_data[idx]['data'])
        col.set_title(plot_data[idx]['title'])
        col.set_ylim([0,1])
        col.set_xticks(range(1,11))
        col.set_xticklabels(range(10))
        col.set_xlabel('label')
        col.set_ylabel('cosine similarity')
        
    plt.show()
    plt.close()

num_img = 1000
img = x_test_norm[:num_img]
img_label = y_test[:num_img]
for result in training_results:
    hyperparam = result['hyperparam']
    print(hyperparam)
    model = result['results']['model']
    encoded_img = model.encoder(img).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    similarity_plot(img=norm_encoded_img, ls_label=img_label)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}

9. Latent space in T-SNE space

def tsne_plot(
    space_gp: List[np.ndarray],
    label: List[int]
) -> None:
    
    _, ax = plt.subplots(nrows=1, ncols=2, figsize=(20,10))
    for idx, col in enumerate(ax):
        space = space_gp[idx]['space']
        scatter = col.scatter(space[:,0], space[:,1], c=label, cmap='Spectral')
        col.set_title(space_gp[idx]['title'])
        col.set_xlabel('tsne 1')
        col.set_ylabel('tsne 2')
        col.legend(*scatter.legend_elements())
    plt.show()
    plt.close()

for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    print('fitting tsne for encoded_img')
    tsne_space = TSNE(n_components=2, n_jobs=-1).fit_transform(encoded_img)
    print('fitting tsne for norm_encoded_img')
    tsne_space_norm = TSNE(n_components=2, n_jobs=-1).fit_transform(norm_encoded_img)
    space_gp = [
        {
            'space': tsne_space,
            'title': 'Latest space',
        },
        {
            'space': tsne_space_norm,
            'title': 'Normalized latest space',
        },
    ]
    tsne_plot(space_gp=space_gp, label=y_test)

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
fitting tsne for encoded_img
fitting tsne for norm_encoded_img

10. K-Mean plot

def kmean_plot(
    img: np.ndarray,
    max_n_cluster: int = 10,
) -> None:
    
    _, ax = plt.subplots(nrows=2, ncols=5, figsize=(25,12))
    ax = ax.flatten()
    range_clusters = range(2,max_n_cluster+1)
    for n_clusters in range_clusters:
        kmeans = KMeans(n_clusters=n_clusters)
        kmeans.fit(img)
        labels = kmeans.labels_
        avg_silhouette_score = silhouette_score(img, labels, metric='euclidean')
        sample_silhouette_values = silhouette_samples(img, labels)
        y_lower = 10
        for i in range(n_clusters):
            ith_cluster_silhouette_values = sample_silhouette_values[labels == i]
            ith_cluster_silhouette_values.sort()
            size_cluster_i = ith_cluster_silhouette_values.shape[0]
            y_upper = y_lower + size_cluster_i
            color = cm.nipy_spectral(float(i) / n_clusters)
            ax[n_clusters-1].fill_betweenx(
                np.arange(y_lower, y_upper),
                0, ith_cluster_silhouette_values,
                facecolor=color, 
                edgecolor=color, 
                alpha=0.7
            )
            ax[n_clusters-1].set_xlim([-0.1, 1])
            ax[n_clusters-1].set_ylim([0, len(img) + (n_clusters + 1) * 10])
            ax[n_clusters-1].text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
            ax[n_clusters-1].set_xlabel('silhouette score')
            ax[n_clusters-1].set_ylabel('cluster label')
            ax[n_clusters-1].set_title(f'{n_clusters}-clusters')
            ax[n_clusters-1].axvline(x=avg_silhouette_score, color="red", linestyle="--")
            ax[n_clusters-1].set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
            ax[n_clusters-1].set_yticks([])
            y_lower = y_upper + 10

    plt.show()
    plt.close()

ls_inertia = list()
for result in training_results:
    model = result['results']['model']
    hyperparam = result['hyperparam']
    print(hyperparam)
    encoded_img = model.encoder(x_test_norm).numpy()
    norm_encoded_img = np.divide(encoded_img,np.linalg.norm(encoded_img, axis=1).reshape(-1,1))
    space_gp = [
        {
            'space': encoded_img,
            'title': 'Latest space',
        },
        {
            'space': norm_encoded_img,
            'title': 'Normalized latest space',
        },
    ]
    for space in space_gp:
        print(f'img = {space["title"]}')
        kmean_plot(img=space['space'])

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 1.1111111111111112e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 2.2222222222222223e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 3.3333333333333335e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 4.4444444444444447e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 5.555555555555556e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 6.666666666666667e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 7.777777777777778e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 8.888888888888889e-05}
img = Latest space

img = Normalized latest space

{'activity_regularizer': 0.0, 'kernel_regularizer': 0.0001}
img = Latest space

img = Normalized latest space